Parametric Equation of a Hyperbola

I. Introduction to Parametric Equations:

• Parametric equations express the coordinates of a point in terms of one or more parameters.
• In the context of conic sections, parametric equations are particularly useful for describing the motion of a point along a curve.

II. Parametric Equation of a Hyperbola: For a hyperbola, the parametric equations are typically expressed in terms of trigonometric functions. The standard parametric equations for a hyperbola centered at the origin are:

$x\left(t\right)=a\mathrm{sec}\left(t\right)$ $y\left(t\right)=b\mathrm{tan}\left(t\right)$

• $\left(x\left(t\right),y\left(t\right)\right)$ represents a point on the hyperbola.
• $t$ is the parameter that varies, and it usually ranges over the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ or $\left(0,2\pi \right)$.

III. Key Parameters:

• $a$: Semi-major axis, distance from the center to the vertices along the x-axis.
• $b$: Semi-minor axis, distance from the center to the vertices along the y-axis.

IV. Graphical Interpretation:

• As the parameter $t$ varies, the parametric equations generate points on the hyperbola.
• The parametric equations describe the relationship between $x$ and $y$ in a dynamic way, capturing the entire shape of the hyperbola as $t$ varies.

Example :

Suppose we have the parametric equations:

$x\left(t\right)=3\mathrm{sec}\left(t\right)$

$y\left(t\right)=2\mathrm{tan}\left(t\right)$

1. Identify Key Parameters:

• Semi-Major Axis ($a$): $a=3$ (coefficient of $\mathrm{sec}\left(t\right)$)
• Semi-Minor Axis ($b$): $b=2$ (coefficient of $\mathrm{tan}\left(t\right)$)

2. Eccentricity ($e$):

• Use the formula $e=\sqrt{1+\frac{{b}^{2}}{{a}^{2}}}$ to find eccentricity.
• $e=\sqrt{1+\frac{{2}^{2}}{{3}^{2}}}=\sqrt{\frac{13}{9}}$

3. Foci ($\left({F}_{1},{F}_{2}\right)$):

• Use the formula $c=\sqrt{{a}^{2}+{b}^{2}}$  to find the distance from the center to each focus.
• $c=\sqrt{{3}^{2}+{2}^{2}}=\sqrt{13}$
• Foci are located at $\left(±\sqrt{13},0\right)$.